SOLUTION: Use reference angles to find the exact value of the expression. Do not use a calculator: csc660 degrees csc660= 1/sin 660 I can find the answer when using the calculator, bu

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Question 884444: Use reference angles to find the exact value of the expression. Do not use a calculator: csc660 degrees
csc660= 1/sin 660
I can find the answer when using the calculator, but I am lost as to figuring out the next step when not using the calc.
Thank you in advance!
Found 2 solutions by josmiceli, rothauserc:
Answer by josmiceli(19441) (Show Source):
You can put this solution on YOUR website!
If you rotate the unit vector CCW twice around
starting at 0 degrees, you have rotated
720 degrees in the positive direction
and are back where you started.
---------------------------
Now if you back off ( CW direction ) by
60 degrees you are at 660 degrees.
---------------------------------
the csc of 660 degrees ( 4th quadrant )
is
-------------------
With calculator:

Answer by rothauserc(4718) (Show Source):
You can put this solution on YOUR website!
csc660= 1/sin 660
note that 660 - 360 = 300
300 degrees is in the fourth quadrant and its reference angle is 360 - 300 = 60 degrees
note that the sine function in quadrant 4 is negative
csc660 = 1/sin 660 = 1 / -sin 60
now use a right triangle with hypotenuse 2, altitude 1 and base is square root of 3
60 degree angle is opposite the base
therefore the sin 60 = square root (3) / 2
now we can write
csc660 = 1 / (-square root (3) / 2) = 2 / -square root(3)
multiply the numerator and denominator by square root(3)
csc660 = (-2*square root(3)) / 3